Relating incompatibility, noncommutativity, uncertainty and Kirkwood-Dirac nonclassicality

TL;DR

This paper explores the relationships between incompatibility, noncommutativity, uncertainty, and Kirkwood-Dirac nonclassicality in quantum states, providing bounds and characterizations relevant to quantum information and metrology.

Contribution

It establishes general bounds linking support uncertainty to Kirkwood-Dirac nonclassicality and characterizes classical states for nearly mutually unbiased observables.

Findings

Support uncertainty serves as an effective Kirkwood-Dirac nonclassicality witness.

Classical states are eigenvectors of incompatible observables close to mutually unbiased.

Complete incompatibility entails several weaker incompatibility notions and strong noncommutativity.

Abstract

We provide an in-depth study of the recently introduced notion of completely incompatible observables and its links to the support uncertainty and to the Kirkwood-Dirac nonclassicality of pure quantum states. The latter notion has recently been proven central to a number of issues in quantum information theory and quantum metrology. In this last context, it was shown that a quantum advantage requires the use of Kirkwood-Dirac nonclassical states. We establish sharp bounds of very general validity that imply that the support uncertainty is an efficient Kirkwood-Dirac nonclassicality witness. When adapted to completely incompatible observables that are close to mutually unbiased ones, this bound allows us to fully characterize the Kirkwood-Dirac classical states as the eigenvectors of the two observables. We show furthermore that complete incompatibility implies several weaker notions of incompatibility, among which features a strong form of noncommutativity.